# Properties

 Label 66270.k Number of curves 8 Conductor 66270 CM no Rank 0 Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("66270.k1")

sage: E.isogeny_class()

## Elliptic curves in class 66270.k

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
66270.k1 66270m7 [1, 0, 1, -11781748, 15564503978] [2] 2543616
66270.k2 66270m8 [1, 0, 1, -1001828, 52446506] [2] 2543616
66270.k3 66270m6 [1, 0, 1, -736748, 242879978] [2, 2] 1271808
66270.k4 66270m5 [1, 0, 1, -637343, -195893692] [2] 847872
66270.k5 66270m4 [1, 0, 1, -151363, 19510316] [2] 847872
66270.k6 66270m2 [1, 0, 1, -40913, -2888944] [2, 2] 423936
66270.k7 66270m3 [1, 0, 1, -29868, 6499306] [2] 635904
66270.k8 66270m1 [1, 0, 1, 3267, -220472] [2] 211968 $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 66270.k have rank $$0$$.

## Modular form 66270.2.a.k

sage: E.q_eigenform(10)

$$q - q^{2} + q^{3} + q^{4} + q^{5} - q^{6} - 4q^{7} - q^{8} + q^{9} - q^{10} + q^{12} - 2q^{13} + 4q^{14} + q^{15} + q^{16} + 6q^{17} - q^{18} + 4q^{19} + O(q^{20})$$

## Isogeny matrix

sage: E.isogeny_class().matrix()

The $$i,j$$ entry is the smallest degree of a cyclic isogeny between the $$i$$-th and $$j$$-th curve in the isogeny class, in the LMFDB numbering.

$$\left(\begin{array}{rrrrrrrr} 1 & 4 & 2 & 12 & 3 & 6 & 4 & 12 \\ 4 & 1 & 2 & 3 & 12 & 6 & 4 & 12 \\ 2 & 2 & 1 & 6 & 6 & 3 & 2 & 6 \\ 12 & 3 & 6 & 1 & 4 & 2 & 12 & 4 \\ 3 & 12 & 6 & 4 & 1 & 2 & 12 & 4 \\ 6 & 6 & 3 & 2 & 2 & 1 & 6 & 2 \\ 4 & 4 & 2 & 12 & 12 & 6 & 1 & 3 \\ 12 & 12 & 6 & 4 & 4 & 2 & 3 & 1 \end{array}\right)$$

## Isogeny graph

sage: E.isogeny_graph().plot(edge_labels=True)

The vertices are labelled with LMFDB labels.